Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×C4×C4⋊C4 ⋊ C2 |
| Sylow | C3 ⋊ #2452 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×C4×C4⋊C4 | C2 |
| prime radical | C22×D12 | C22 |
| hypercenter | C4×C4⋊C4 | D3 |
| hypocenter | C3 | #2452 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 | |
620 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(8,10,11,9)(12,14,15,19)(13,17,16,18);(5,7)(8,9,11,10);(4,5)(6,7)(8,9,11,10)(13,16)(17,18);(2,3)(9,10)(12,13,15,16)(14,18,19,17)Polycyclic presentation:
2^7,3|a^2=e,b^2=f,c^2=f,d^2=ef,b^a=bef,c^a=cf,d^a=def,h^a=h^2,c^b=cg78 conjugacy classes — view the full character table.
| maximal subgroup of | #10858138, #10858143, #10858213, … |
| maximal normal subgroup of | #10858138, #10858143, #10858213, … |
| maximal quotient of | #10717210, #10717556, #10718069, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 6251);